CMMA月例セミナー
CMMA月例セミナー
主催  共同利用・共同研究拠点 明治大学先端数理科学インスティテュート(MIMS) 「現象数理学拠点」 

中野キャンパス へのアクセス 

中野キャンパス マップ 
講演は日本語で行われます。 The seminar will be lectured in Japanese.
第22回CMMA月例セミナー
 日時 :
 2018年1月25日（木） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 俣野博 (Hiroshi Matano)
(明治大学/ MIMS所員，東京大学)

Title：
"Front propagation in
nonlinear diffusion equations on
the hyperbolic space"
 Abstract：
In this talk I will consider reactiondiffusion equations on the hyperbolic space H n. Unlike the Euclidean space, which is flat, the hyperbolic space is a space that is warped with constant negative curvature. Our goal is to understand the effect of the curvature on the behavior of solutions. More specifically we consider solutions with localized initial data, and study how the solution fronts spread over the space; in particular, we will discuss how the curvature of the space affects the spreading speed of the fronts. The talk is focused mainly on scalar reactiondiffusion equations, but I will also discuss briefly predatorprey models on such spaces.
This is partly joint work with Fabio Punzo and Alberto Tesei
第21回CMMA月例セミナー
 日時 :
 2017年11月27日（月） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 412教室
 講師：
 奈良 知惠 (Chie Nara)
(明治大学/ 客員教授，MIMS所員)

Title：
"Foldable Products Based on
Discrete Geometry and Origami Engineering"
 Abstract：
There are many commercial foldable products in our daily life, foldable umbrellas and airbags for cars which are typical examples. The foldability is advantage, both carrying and storage, because of compactness. However, there are still many things which we wish to be foldable. Can we flatten empty containers of juice without cutting or stretching? The answer depends on conditions such as shapes and materials. In this talk, by applying recent results on continuous flattening problems in Discrete Geometry and combining them with the Origami engineering, we introduce some examples of foldable products such as helmets and boxes. Future works will also be discussed.
第20回CMMA月例セミナー
 日時 :
 2017年10月23日（月） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 嵯峨山茂樹 (Shigeki Sagayama)
(明治大学/ 教授，MIMS所員)

Title：
"Stochastic Inverse Problems in
Music Processing"
 Abstract：
Music is great fun. Even greater fun is music processing research. There are various interesting topics and issues waiting to be solved in the music processing area that this talk will address. Being time series information like speech and language, music is a good (or even better) target for sophisticated models and algorithms developed in the field of spoken language processing, although the focus lies on fundamental frequencies in contrast to the spectrum envelope in the case of speech recognition. Therefore, a significant extension of features and models is often required instead of simple adoption of existing tools. In analogy to spoken language, music processing is threefold at least: music signal corresponding to speech signal, music performance corresponding to pronunciation/prosody and music score corresponding to text. There exist automatic generation/synthesis, decomposition, manipulation/modification and analysis issues within each of these three domains as well as conversion/recognition and rendering/synthesis issues between them.
Signal processing issues include automatic music transcription (as the inverse problem of playing music to a score) and its partial problem, multipitch analysis for polyphonic music, sound, tempo and note value identification, rhythm analysis/recognition, music genre classification, musical instrument identification, score following/alignment and note onset detection. There are also numerous nonsignal (i.e., symbolic) information processing issues such as automatic music composition, arrangement, accompaniment, piano/guitar fingering decision, music rendering from score and an automated jazz session. Among these issues are many inverse problems where probabilistic models and statistical inference are quite helpful.
The entire area of music processing looks as vast as that of speech/language processing. In this talk, some examples of approaches to these scientific issues are introduced along with interesting results.
第19回CMMA月例セミナー
 日時 :
 2017年9月25日（月） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 萩原一郎 (Ichiro Hagiwara)
(明治大学/ 特任教授，MIMS所員)

Title：
"Mathematical Sciences for
Revolution of Development Style of Vehicle"
 Abstract：
It took 40～60 months from plan to market of vehicle because a lot of characteristics which are tradeoff between each other must be solved. Mathematical Sciences have contributed for developing of CAD/CAM/CAE on which results, 3D CAD has been placed for Cray model, precise model for test car and CAE for experiments. Thanks of revolution of changing of developing style, the development period of vehicle has been so much reduced to 12～20 months. CAE consists of heatfluid , sound & vibration, collisionstrengthstiffness and optimization. Fortunately author contributed essentially for the subjects except heatfluid. Here the contents of these CAE are explained and what I did for this CAE are introduced. And at last it will be discussed what type of mathematical sciences will be requested in future for vehicle.
第18回CMMA月例セミナー
 日時 :
 2017年7月18日（火） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 二宮広和 (Hirokazu Ninomiya)
(明治大学/ 教授，MIMS所員)

Title：
"Dynamics of waves in excitable media"
 Abstract：
Pattern formation of reactiondiffusion system has been of interest for many years. Many interesting phenomena have been observed experimentally and numerically. However, it is difficult to handle the dynamics of these phenomena mathematical rigorously. To overcome this difficulty, we introduce a simple model of excitable system based on a free boundary problem. In this talk, I will explain the dynamics of solutions of this model. Especially, I will characterize the dynamics in one dimensional case. Then I will explain the existence of traveling waves in two dimensional excitable media and discuss the role of obstacles in the excitable media. One of the roles of obstacles is spontaneous spiral formation depending on the geometry of obstacle.
第17回CMMA月例セミナー
 日時 :
 2017年6月20日（火） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 山口智彦 (Tomohiko Yamaguchi)
(明治大学/ 特任教授，MIMS所員)

Title：
"SelfOrganization and Biomimetic Systems"
 Abstract：
Selforganization is a process through which the order in a system increases without any external control of human beings. Since selforganization is typically observed in living systems, it has been drawing the attention of many researchers. However, the meaning of this term differed among disciplines until the very end of the last century when the tacit knowledge of selforganization was disclosed from the viewpoint of thermodynamics.
In my talk I will briefly overview a historical background of selforganization, and will introduce biomimetic chemical systems (the BZ reaction in laboratory experiments and the GrayScott model in silico) whose dynamics are well described by PDEs. My talk will be closed by referring to how to extract a quantity of time evolution in terms of organizations and collapses in autonomous open systems.
第16回CMMA月例セミナー
 日時 :
 2017年5月16日（火） 16:30~17:30
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 Elliott GINDER
(明治大学/ 准教授，MIMS研究員)

Title：
"Multiphase optimization in
phononic crystal design"
 Abstract：
This research approaches the design and imaging of phononic crystals (PnC) through means of experimentation, mathematics, and computation. We will present surface wave imaging results of composite elastic materials where we are aiming at the development of techniques for preforming noninvasive CT imaging. Finite element methods for approximating the solution to the model equations are then used to investigate the control of bandgaps through related eigenvalue problems. We will also remark about our technique for expressing the multiphase nature of PnC and its role in formulating shape and density gradient optimization problems.
第15回CMMA月例セミナー
 日時 :
 2016年5月16日（月） 17:20~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 井倉弓彦 (Yumihiko Ikura)
(明治大学/ 特任講師，MIMS研究員)

Title：
"The surface pattern that
depends on the shape of
swelling hydrogel"
 Abstract：
In this topic, I will introduce our study on the mechanism of a crease pattern on a hydorogel in the process of swelling. Most studies on an instability of the hydrogel swelling have been considered with constraint boundary condition. In our study, we investigated the mechanical instabilities of a discshaped thin plate of polyelectrolyte hydrogel with fast free swelling. In the lead up to an equilibrium swelling, the gel morphed into saddlelike shape with a polygonal surface due to two swelling mismatches, inner disc/outer annulus region, and surface layer/inner layers. And then, the polygonal surface crease pattern gradually changed to stripelike pattern. The stripe creases were formed unidirectionally, and were normal to each other at the top and bottom surfaces. From the detail observation of an internal stress analyzed by the orientation of semirigid polyanions, we considered that both the surface creasing and bulk deformation minimize the swelling mismatch. And a theoretical energy model [1] was developed to show anisotropic stripe crease on a saddle shaped surface. These results might be helpful to develop novel strategies for controlling crease patterns on soft and wet materials by changing its threedimensional shape.
[1] T. Tanaka, et al, Nature, 1987, 325, 796.
第14回CMMA月例セミナー
 日時 :
 2016年5月16日（月） 16:30~17:10
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 近藤信太郎 (Shintaro Kondo)
(明治大学/ MIMS研究員)

Title：
"A macroscopic model for
understanding lightness
optical illusions"
 Abstract：
There are two type lightness optical illusions, one is to emphasize the contrast of brightness (brightness contrast) and the other is to weaken the contrast of brightness (brightness assimilation). In particular, it is considered that brightness contrast exists to capture clearly the contour of the object, and it is caused by lateral inhibition in retina. Therefore many visual models assume the function of lateral inhibition. Our new macroscopic model does not assume the lateral inhibition (i.e., does not assume the function to emphasize the contours of the object), and mimic a normal information processing in the retina. We find that our model can reproduce the contrast and assimilation simultaneously. This suggests that in retina there is no function to emphasize the contour of the object aggressively. Moreover, we find that our model can't represent the contrast if the parameters fulfill certain conditions. This is likely to represent the individual differences of information processing in the retina.
第13回CMMA月例セミナー
 日時 :
 2016年4月12日（火） 16:20~17:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 関坂歩幹 (Ayuki Sekisaka)
(明治大学/ MIMS研究員)

Title：
"The absolute spectrum and
the essential spectrum"
 Abstract：
We interest in the stability of traveling waves for onedimensional reactiondiffusion systems.Stability of waves is given by the spectrum associated with the linearized operator.
The existence and stability of traveling waves are typically investigated on unbounded intervals even though the domains in experiments or numerical simulations are bounded. Instabilities of waves on unbounded intervals can manifest themselves in different ways. One is the socalled absolute instability and another is the socalled convective instability related to the discrete spectrum and the essential spectrum, respectively. Except possibly for finitely many eigenvalues that are created by the boundary condition or the special dependence of waves, all eigenvalues of waves on the large bounded interval are close to the absolute spectrum. The spectrum of waves on the unbounded interval has the essential spectrum but the bounded interval case has only the discrete spectrum. Such a difference produces a difference in stability of waves related to the size of interval because a location of the absolute spectrum does not accord with the essential spectrum.
In this talk, we introduce two instabilities of waves via a simple example, and show the topological mechanism of the accumulation of eigenvalues. Furthermore, we show that these problems appear in the gluing bifurcation problem of traveling waves.
第12回CMMA月例セミナー
 日時 :
 2016年4月12日（火） 15:40~16:20
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 Nina Vladimirovna Sviridova
(明治大学/ MIMS研究員)

Title：
"Study on
global and local chaotic characteristics
of human photoplethysmogram"
 Abstract：
Many of widely used quantitative nonlinear time series analysis methods provide results corresponding to data dynamics overall, i.e. obtained by taking into account all reconstructed attractor. Such results can describe dynamical properties in “global” scale, however achievement of better understanding of phenomenon itself, as well as attractor trajectory evolving process may require investigation of local (on a fixed area on attractor) characteristics. Thus in a case of dealing with, for example, biological data, local areas on the reconstructed attractor may refer to the phases of the process, therefore local characteristics, unlike global one, may describe and be able to differentiate each phase. Investigation of local characteristics may also provide deeper insight on interaction between different processes or effects of noise on chaotic dynamics. The objectives of my previous research were to investigate “global” chaotic characteristics of human photoplethysmogram and prove that its dynamics consistent with chaotic motion definition, as well as to study local predictability and noise effects on chaotic dynamics by local analysis.
Results of “global” properties study demonstrated that photoplethysmogram dynamics is consistent with the definition of chaotic movement. It was also found that deterministic nonlinear prediction, Poincare section and the Wayland test can reveal important characteristics of photoplethysmographic signals that is important for theoretical and applied studies on the photoplethysmogram. “Local” analysis applied to wellknown chaotic models revealed presence of local noise sensitivity, as well as local orbital instability. Future study aims to apply concept of local noise sensitivity and orbital instability to human photoplethysmogram and electrocardiogram and investigate appropriateness of these local characteristics usage for application studies.
第11回CMMA月例セミナー
 日時 :
 2016年4月12日（火） 15:00~15:40
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 Contento Lorenzo
(明治大学/ MIMS研究員)

Title：
"Traveling waves in a threespecies
competitiondiffusion system"
 Abstract：
From the ecological point of view of competitormediated coexistence, we consider a threespecies competitiondiffusion system which models one exotic competing species W invading the native system of two strongly competing species U and V. Even if W is weaker than the native species, both competitive exclusion and competitormediated coexistence may occur, depending on r3, the growth rate of W. This phenomenon is deeply linked with the existence of two onedimensional stable traveling waves involving the species (U,V) and (U,V,W ), respectively, and whose relative speed depends on r3. Studying the interaction of these waves offers insight about the patterns appearing in two dimensions. In particular, the transition from regular patterns to complex ones is deeply linked to the destabilization of a threespecies traveling pulse and the appearance of breathing traveling waves. In two spatial dimensions the interaction of nonparallel planar fronts generates instead twodimensional traveling waves, whose shape depends on r3 and can be of wedge, zipper or biwedge type. In our future work we intend to study similarly shaped waves in a simpler twoactivator oneinhibitor FitzHughNagumo system and to improve our understanding of the conditions for species coexistence in the competitiondiffusion system.
第10回CMMA月例セミナー
 日時 :
 2016年1月13日（水） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 須志田隆道 (Takamichi Sushida)
(明治大学/ MIMS研究員)

Title：
"A reactiondiffusion model for
understanding optical illusions"
 Abstract：
Optical illusion is the phenomenon in which what we perceive is different from physical reality. It is automatically induced regardless of our consciousness. Thus optical illusion is a natural phenomenon in our visual perception. In our visual processing, it is known that a brightness stimulus is converted to an electrical signal by passing through several processes. It is an important property that brightness stimulus is emphasized and controlled in its processes. In this study, we assume that the blur effect, the contrast effect, and the assimilation effect operate to brightness stimulus in our visual processing. The blur effect corresponds to a conversion degree of an electrical signal in the photoreceptor cell for brightness stimulus received in the retina. The contrast effect plays a role which emphasize difference in its electrical signal, and the assimilation effect plays its reversal effect. In the spatial frequency characteristics, it is known that high spatial frequency induces an assimilation phenomenon and low spatial frequency induces a contrast phenomenon. That is, it is considered that superiority of its two effects switch for converted brightness stimulus. In this presentation, we propose a mathematical model assuming the above three effects. By mathematical analysis, we derive an integral equation for timeindependent problem and show a necessary and sufficient condition that its integral kernel is a function type called Mexicanhat. Moreover, we discuss perceptions of optical illusions such as the spatial frequency characteristics by numerical simulations.
第9回CMMA月例セミナー
 日時 :
 2015年12月14日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 岩本真裕子 (Mayuko Iwamoto) (明治大学/ MIMS研究員)

Title：
"A simple model for
proportion regulation in crowds"
 Abstract：
Selforganizing orderly systems are often observed in nature. One of the fundamental mysteries in biology concerns the method by which a cluster of organisms can regulate the proportion of individuals that perform various roles as if each individual knows a whole situation, without a leader. That means a question how flexibility in resulting order system can be realized by local interaction only. In this presentation, we focus on the proportion regulation between two kinds of modes, and propose the simple model as a basis of the mechanism by a series of local interactions between individuals. Our numerical simulations with cellular automata model indicate that asymmetric interaction and discrete system are crucial to response to global situations.
第8回CMMA月例セミナー
 日時 :
 2015年11月30日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 物部治徳 (Harunori Monobe) (明治大学/ MIMS研究員)

Title：
"Mathematical analysis for free boundary and
interface problems arising from
ice melting and cell crawling"
 Abstract：
Interfaces are appeared in various physical and biological phenomena and most of the interfaces change a shape momentarily. For instance, when the ice is melting, the shape of ice changes. Also, when cells crawl, the shape of cell membrane changes. To understand such phenomena, some partial differential equations, called free boundary and interface problems, are proposed. In this talk, I introduce some results of free boundary and interface problems related to ice melting and cell crawling etc. In particular, I mainly would like to talk about ''Reactiondiffusion system approximation :A Singular limit problems of some Reaction diffusion systems'' and ''Existence of traveling wave solutions of free boundary and interface problems''.
第7回CMMA月例セミナー
 日時 :
 2015年9月11日（金） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 杉原厚吉 (Kokichi Sugihara) (明治大学/ MIMS所員)

Title：
"Fragility of Monocular Vision
Suggested by a Mathematical Model of
Human Depth Illusion"
 Abstract：
There is great difference between the real world and its images; the real world is threedimensional whereas the images are twodimensional. In spite of this difference we can enjoy visual media without any special efforts. Why can we do this? This question can be partly answered by mathematical study of depth illusion. It seems that the human brains try to recover the depth from images using strong performances for some special subclass of objects such as rectangular solids. This observation also suggests fragility of the visual media culture. We discuss this point using optical illusion raised by unusual solids including impossible objects and ambiguous objects.
第6回CMMA月例セミナー
 日時 :
 2015年7月13日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 石田祥子 (Sachiko Ishida) (明治大学/ MIMS研究員)

Title：
"Approach to Design of Mechanical Devices
Using Foldable Structures"
 Abstract：
In this talk, the mechanical properties of the foldable structures and the appropriate application to mechanical devices are introduced. The foldable designs that enable to change the shapes dramatically have been mainly applied to mechanical devices including space structures, architecture, robots, and medical devices to make them compact and portable. However, remarkable properties of the foldable structures are not only the compactness and portability. Thus, numerical computation was performed on each foldable design to disclose the further properties such as nondeformative motions, amplification mechanism and bistability of the shapes, and to propose the appropriate application to mechanical devices utilizing the properties such as pneumatic actuators, displacement amplifiers, and vibration isolators.
第5回CMMA月例セミナー
 日時 :
 2015年5月25日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 池田幸太 (Kota Ikeda) (明治大学/ MIMS所員)
Title：
"Congestion flow of pulses in
an excitable reactiondiffusion system"
 Abstract：
Selfdriven motion is observed in several fields, e.g., biology, chemistry, and nonlinear physics. Organisms move spontaneously to aggregate and form selforganized structures. As a spatiotemporal collective motion, congestion flow is observed in a system with animal and inanimal organisms. For example, camphor boats constitute a system for changing the number of particles and with simple interaction and generate congestion flow as reported by Suematsu et al in 2010. The mechanism of the congestion dynamics of camphor boats has been investigated theoretically. As stated in our previous works, a traveling wave solution in a model with an inhomogeneity plays an important role.
Recently it was reported that traveling wave solutions with a pulse shape, simply called traveling pulses, could generate congestion flow in a reactiondiffusion system with excitability. It is wellknown that a traveling pulse is formed spontaneously in an excitable system like the FitzHughNagumo model. This fact seems to imply that the same mechanism as in a system with camphor boats works in the congestion flow of an excitable system. However, it is not true because the reactiondiffusion system has no inhomogeneity. In this talk, we focus on studying the traveling pulse and consider what is different between the congestion flow in the model of camphor boats and the excitable system.
第4回CMMA月例セミナー
 日時 :
 2015年4月20日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 荒川薫 (Kaoru Arakawa) (明治大学/ MIMS所員)
Title：
"A Method of Color Scheme with
High Contrast Using
Interactive Evolutionary Computing"
 Abstract：
A method of color scheme for graphical images is proposed using interactive evolutionary computing. This method can set the contrast of luminance between adjacent regions high, while setting preferable color, considering the user’s subjective criteria. Keeping the contrast of luminance high, this system enables users with visual disturbance to identify each region in graphical image clearly. As to the color, samples of wellknown fine color combinations proposed by artists are adopted, so that the obtained color design is fine and harmonic. In order to design both the luminance and color totally considering human subjective criteria, interactive evolutionary computing is applied. Here, the luminance, contrast and color in each region are represented as individuals in the genetic algorithm and are updated, so that the output design is preferable to the user. Computer simulations verify the high performance of this system.
第3回CMMA月例セミナー
 日時 :
 2015年1月16日（金） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 青木健一 (Kenichi Aoki) (明治大学/ MIMS研究員)
Title：
"Modeling 2,600,000 Years of
Human Cultural Evolution"
 Abstract：
Cultural evolution during the Palaeolithic/Stone Age was apparently characterized by long periods of stasis punctuated by abrupt regime shifts. In order to interpret this pattern of cultural evolution, the coupled dynamics of the size and cultural/technological level of a population, with positive feedback between these two variables, is modeled in the classical MalthusianBoserupian framework. What is new is that innovativeness is included as a parameter, because genetic changes in cognition cannot be ignored, given the immense span of biologicalevolutionary time involved and the fact that brain size increased at least twofold. Bifurcation diagrams, with innovativeness as the parameter, show that abrupt transitions in the cultural level“catastrophic bifurcations”are possible. This modified MalthusianBoserupian model can also explain the correlations between toolkit size and population size, or the absence thereof, among ethnographic societies. Finally, time permitting I briefly describe a reactiondiffusion model for the geographical spread of early farming in Europe, which accounts for the observed constant rate of advance.
第2回CMMA月例セミナー
 日時 :
 2014年12月15日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 末松 J. 信彦 (Nobuhiko J. Suematsu)
(明治大学/ MIMS所員)
Title：
"Experimental estimation of
the parameters of mathematical model for
selfpropelled camphor boat"
 Abstract：
Selfpropelled systems are actively investigated as a model experimental systems of living organisms and/or a novel type of functional materials. Indeed, several types of selfpropelled systems have been reported and these systems exhibited characteristic behaviors and functions such as taxis. In addition, mathematical approaches have proposed several novel functional systems for selfpropelled systems. To realize these expectations and progress the development of smart selfpropelled systems, quantitative information regarding selfpropelled motion is required.
In this presentation, we focused on a camphor boat as a simple selfpropelled system and quantitatively measured the main parameters for the system. We then confirmed the validity of the mathematical model for camphor motion and the values of the parameters.
第1回CMMA月例セミナー
 日時 :
 2014年11月10日（月） 17:00~18:00
 場所 :
 明治大学 中野キャンパス高層棟6階 研究セミナー室3
 講師：
 宮路智行 (Tomoyuki Miyaji) (明治大学/ MIMS所員)
Title：
"A computerassisted proof of
existence of a periodic orbit"
 Abstract：
We consider a threedimensional dynamical system defined by ordinary differential equations proposed by Craik and Okamoto. It arises in the context of the fluid dynamics. It is untypical in that most of the solution orbits are unbounded. It is conjectured that an unstable periodic orbit determines behaviour of unbounded orbits which draw a helical curve. We have proved the existence and local uniqueness of the periodic orbit with the aid of a computer. In this talk, we introduce the method of computerassisted proof, and show some recent progress on the dynamical system.